Apply mathematics to problems arising in everyday life, society, and the workplace

Standards Alignment

• TEKS.111.28.01A primary
  apply mathematics to problems arising in everyday life, society, and the workplace;
• TEKS.111.28.01B primary
  use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;
• TEKS.111.28.01C primary
  select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

Lesson Overview

This lesson engages Grade 8 students in applying mathematics to solve problems arising in everyday life, society, and the workplace. Students will use a structured problem-solving model that includes analyzing information, planning, solving, justifying, and evaluating solutions. They will select appropriate tools and techniques such as manipulatives, drawings, mental math, and technology to support their problem-solving process. The lesson emphasizes understanding strategies and reasoning, connecting math skills to real-world contexts, and reflecting on the reasonableness of solutions.

Learning Objectives

• Apply mathematics to solve problems that arise in everyday life, society, and the workplace.
• Use a problem-solving model to analyze information, plan a strategy, determine a solution, justify the solution, and evaluate the problem-solving process and the reasonableness of the solution.
• Select and use appropriate tools and techniques, including manipulatives, drawings, paper and pencil, technology, mental math, estimation, and number sense, to solve problems.

Success Criteria

• Students can identify and analyze given information in a real-world problem.
• Students can formulate a plan or strategy to solve the problem using appropriate mathematical tools.
• Students can determine and justify a solution to the problem.
• Students can evaluate the problem-solving process and explain whether the solution is reasonable.
• Students can explain their problem-solving strategies clearly to peers.

Prerequisite Knowledge

Students should be familiar with basic arithmetic operations, simple algebraic expressions, and have experience using manipulatives or drawings to represent mathematical ideas.

Key Vocabulary

• Problem-solving model
• Analyze
• Plan/Strategy
• Solution
• Justify
• Evaluate
• Manipulatives
• Estimation
• Number sense
• Reasonableness

Materials and Resources

• Concrete materials or manipulatives (e.g., counters, blocks)
• Paper and pencil
• Graphing calculators or computers/tablets with math software (optional)
• Whiteboard or chart paper for modeling
• Real-world problem scenarios handout

Teacher Preparation

• Prepare one or two short real-world problems relevant to students' daily life or workplace contexts.
• Gather manipulatives and drawing materials for modeling.
• Prepare guiding questions to prompt student thinking during problem-solving.
• Set up technology tools if available for student use.

Detailed Lesson Notes

Understanding the Problem-Solving Model

The problem-solving model used in this lesson involves five key steps: analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the process and the reasonableness of the solution. This structured approach helps students break down complex problems into manageable parts and develop logical reasoning skills.

Applying Mathematics to Real-World Problems

Mathematics is a tool to solve problems encountered in everyday life, society, and the workplace. Examples include calculating expenses, comparing prices, budgeting, or determining travel time. Applying math in these contexts requires understanding the problem, selecting appropriate mathematical operations, and interpreting results in context.

Selecting Appropriate Tools and Techniques

Students should learn to choose tools that best support their problem-solving. Concrete objects or manipulatives help visualize problems, drawings can represent relationships, and paper and pencil allow for calculations and notes. Technology can assist with complex computations or graphing. Techniques like mental math and estimation help check answers for reasonableness and efficiency.

Modeling with Concrete Materials and Drawings

Using manipulatives or drawings makes abstract problems tangible. For example, to solve a problem about sharing items equally, counters can represent the items, and students can physically divide them. Drawing diagrams or number lines can illustrate relationships or steps in a problem, aiding comprehension and communication.

Justifying and Evaluating Solutions

Students must explain why their solution makes sense and how they arrived at it. Justification involves articulating reasoning and mathematical operations used. Evaluation includes checking calculations, considering alternative strategies, and reflecting on whether the solution fits the problem context and is reasonable.

Worked Examples

Classroom Activity

Work collaboratively to use manipulatives or drawings to represent the problem, decide on a plan, and solve it.

Guided Practice

Independent Practice

1. Foundational: Identify the important information and what needs to be found in this problem: "You have $20 and want to buy a book that costs $15. How much money will you have left after the purchase?"
2. Developing: Using paper and pencil, calculate the total cost if you buy 4 sandwiches at $3 each and a drink for $2.50. What is the total amount you will pay?
3. Application: Explain how you would use a drawing or manipulatives to solve this problem: "You have 12 apples and want to share them equally among 4 friends. How many apples does each friend get?"
4. Analysis: A store offers a 10% discount on a $50 jacket. Describe the steps you would take to find the sale price and explain why your solution is reasonable.
5. Challenge: You want to buy 3 items priced $7.25, $4.50, and $6.80. You have a $25 bill. Use estimation to decide if you have enough money before calculating the exact total.

Independent Practice Teacher Answer Key

1. 1. Important information: You have $20, the book costs $15. Need to find: How much money is left after buying the book.
2. 2. Multiply 4 × $3 = $12, then add $2.50. Total cost = $12 + $2.50 = $14.50.
3. 3. You could draw 12 apples and divide them into 4 groups, or use 12 counters and physically separate them into 4 equal piles. Each pile will have 3 apples.
4. 4. First, find 10% of $50 by multiplying 0.10 × $50 = $5. Then subtract $5 from $50 to get $45. This is reasonable because a 10% discount reduces the price by one-tenth.
5. 5. Estimate by rounding prices: $7.25 ≈ $7, $4.50 ≈ $5, $6.80 ≈ $7. Sum estimated prices: $7 + $5 + $7 = $19, which is less than $25, so likely enough money. Then calculate exact total: $7.25 + $4.50 + $6.80 = $18.55, confirming enough money.

Guiding Questions

• What information is important in this problem?
• How can we represent the problem to understand it better?
• What strategy will help us find the solution?
• How do we know our solution is correct?
• Is our answer reasonable in this context?

Common Misconceptions

• Students may focus on finding an answer quickly without planning or justifying their solution.
• Students might not evaluate whether their answer makes sense in the problem context.
• Students may rely solely on one tool or technique and not consider others that might be more efficient.
• Students might confuse the steps of the problem-solving model or skip steps like justification and evaluation.

Differentiation

Assessment

• Observe student participation during guided practice and discussions.
• Ask students to explain their problem-solving steps aloud.
• Use exit tickets with a short real-world problem requiring justification of the solution.
• Solve a real-world problem using the problem-solving model and write a brief explanation of your solution and why it is reasonable.
• Assign a project where students identify a real-life problem, apply mathematics to solve it, and present their solution and reasoning.

Answer Guide

• Identify the important information and what needs to be found in a given problem.
  Answer: Students correctly identify key numbers and the question asked in the problem.
• Calculate total cost using multiplication and addition.
  Answer: Students correctly multiply and add to find the total cost.
• Explain how to use drawings or manipulatives to solve a division problem.
  Answer: Students describe dividing objects or drawings into equal groups to find the answer.
• Describe steps to find a discount and justify the solution.
  Answer: Students explain calculating the percentage of the price and subtracting it, with reasoning.
• Use estimation to decide if money is sufficient before exact calculation.
  Answer: Students round prices to estimate total and compare to available money, then calculate exact total to confirm.

Real-Life Application

Encourage students to identify and solve a math problem at home, such as budgeting for groceries or calculating time for a trip, and share their problem-solving process with family members.

Homework or Home Connection

• Encourage students to identify and solve a math problem at home, such as budgeting for groceries or calculating time for a trip, and share their problem-solving process with family members.

Lesson Summary

In this lesson, students learn to apply mathematics to real-life problems by using a structured problem-solving model. They practice analyzing information, planning strategies, selecting appropriate tools, solving problems, justifying their answers, and evaluating the reasonableness of solutions. Through modeling, guided practice, and reflection, students develop skills to confidently use math in everyday life, society, and the workplace.

Teacher Notes

Use the exact standards alignment and retrieved-source provenance stored with this enrichment.

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